You know I have trouble letting stuff go, right? I am still thinking about these crazy long basketball shots. Here are some more thoughts.
Really, there are two things I am interested in. First, commenter Scott Post suggests that the drag coefficient might be around 0.25 instead of 0.5. I don’t know. For the discussion before, it doesn’t really matter. My point was to see a numerical model for a falling ball would be similar to the time and distance from the video. Changing the drag coefficient to 0.25 gives values that are still close to the video. So, I still think the video is real.
The other thing that I would like to explore is the range of launch angles and velocities for a person throwing a ball. This would be important to have a feel for the spread of initial values so that I could estimate how many times you would have to try a shot to make it.
After a short search online, I found two interesting articles. One of them I know you can read – the other you might only be able to access if you are at an academic institution or if you are a subscriber to the American Journal of Physics. Below I will extract the useful details from these two papers.
This pdf paper does not appear to be published in a journal, just online. The author looked at a few shots of a basketball indoors from the same distance away from the goal (5 shots). It seems the main point was to determine if you would need to include air resistance and spin effects in the analysis of this shot. To do this, Saleh used a video camera to capture the 5 shots (one of which was a miss) and then did some video analysis.
For air resistance and spin, the author claims there are not noticeable effects – mostly because there is no horizontal acceleration. It is odd that the vertical values for acceleration were around 9.1 m/s2. But maybe this was just a scaling problem.
Here is a table showing the initial velocities for the 5 shots. This can give some idea of how consistent a shooter could be (although it would be nice to have more than 5 data runs).
That could be useful.
This paper is from the American Journal of Physics (didn’t I already say that?) and really tries to do a lot of stuff. Part of it covers bouncing of a rotating ball (which is interesting, but not what I am looking for). The author also discusses the kinematics of projectile motion (but not with air resistance). There are some more details about projectile motion for an object of finite size that has to hit a target. This, in particular, diagram is interesting:
Basically, the lower the angle of incidence for the basketball, the smaller the apparent size of the hole. And here is another great diagram showing the range of initial velocities and angles that will result in a successful shot (where it doesn’t hit the rim).
Finally, the author compares no-air resistance with air resistance for some cases. The result is that you would have to adjust the initial velocity by about 5% to compensate for the drag.
Where does this leave me?
The problem with both of these papers is that they are for normal basketball shots – not super duper shots, so air resistance isn’t such a big deal. I would really like to have an experimental value for the drag coefficient of a basketball. Probably the best bet would be to get a nice video of a basketball traveling very fast. From this, I should be able to get the coefficient (I will put this on my to-do list).
The second paper was very detailed, but it approached the problem from the wrong direction. It said “what initial conditions would you need to hit the goal”. I want to know what variation a player would have in throwing a ball. Oh well, I think I can get this data myself also.